# Full text of "Treatise On Analysis Vol-Ii"

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```9   EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS       283

(i) It follows from (14.8.1.1) that the function (s9 x)^f(s'1x) is
(M ® jS)-measurable. Moreover, if /? = !, the measure /•/? is bounded
(13.14.4), hence \JL and /are convolvable (14.6.2) and we have

But || /i * (/• p)\\ = NiOi */), which proves (14.9.2.1) in this case. For p = 2
we have, by (13.11.2.2), for any function h e

g ([r

ifc(x)l2d/Kx)dM(s))
also, by (13.21.9) and the left-invariance of jS,

l2 djS(x) d |/i| (s) = |*d M (s) J* lAs-1*)!2 dflx)

from which it follows that \a and /• yS are convolvable. Moreover, by
(13.11.2.2),

/(s" x*)l2 ^ H (0,
so that the above relations and (13.21.9) imply

J

1*)! d|/t|(s))2 ^

which proves (14.9.2.1 ) in this case. Finally, when p = -h oo, we remark that ju
and the function 1 are convolvable by virtue of (14.6.3), and that ju * 1 is
equal to the constant function ju(l). It follows immediately that \JL is con-
volvable with every function in ££%(Q, j3) and that the inequality (14.9.2.1)
is again valid (cf. (14.5.3)).

(ii)   The fact that the integral \f(s~lx) dfj,(s) exists for all x 6 G follows

from (13.20.4). To show that ju */is continuous at a point JCQ e G, we may
limit ourselves to the case where ju ^ 0. For each e > 0, there exists a compact
subset K of G such that ju(()K) ^ e. Let V0 be a compact neighborhood ol
x0. The function / is uniformly continuous on the compact set K~1V0,
hence there exists a neighborhood V c V0 of XQ in G such that the relatior
x e V impliesG, /?). The function u * f (defined almost everywhere by (14.8.2)) then
```